Discrete and continuous models have been proposed to study tumor progression within the cancer stem cells (CSCs) hypothesis. According to this concept, CSCs are a small subpopulation within the tumor that are the main cause for its progression. These cells have high proliferation rate, self-renewal and immortality capacities, moreover, they are very difficult to detect and to eradicate. The objective of our study is to deduce the morphology and fate of growing tumors under the existence of these cells exhibiting a singular behavior. A mathematical growth model is considered in the continuous limit, for a mixed population, having 3 subpopulations (CSCs), Differentiated cells (DCs) and all other cells that a tumor harbors. Analysis and stability of fixed points show the possibility of tumor extinction but also the existence of two stationary states with constant cell concentration indicating that the tumor may reach quiescent states. According to activator and inhibitor concentration which controls the phenotypic change, a phase-diagram is presented which shows the exchange of stability as a function of the physical parameters. Focusing on the conditions for final extinction by a proper choice of the parameters is indeed a crucial goal for efficient therapeutic strategies.

The study is completed by local constraints, interaction between cells and with the environment. However, such study requires to elucidate the pivotal question of cancer stem cell localization. It is possible that the stem-ness properties of cancer cells do not require niches as ordinary stem cells. Nonetheless, the dynamics and patterning of tumors will be strongly affected depending on the existence of niches. Numerical investigation is then necessary varying cell motility during the tumor expansion. In particular, we show that, depending on the stroma, the cell repartition will organize differently.